Stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.

Definition

A complex manifold ${\displaystyle X}$  of complex dimension ${\displaystyle n}$  is called Stein manifold, if the following conditions hold:

• ${\displaystyle X}$  is holomorph convex, i.e. the so-called holomorph convex hull

${\displaystyle {\bar {K}}=\{z,z\in X,|f(z)|\leq sup_{K}|f|forallf\in {\mathcal {O}}(X)\}}$ 

is a compact subset of ${\displaystyle X}$  for every compact subset ${\displaystyle K\subset X}$ . Here ${\displaystyle {\mathcal {O}}(X)}$  denotes the ring of holomorphic functions on ${\displaystyle X}$ .

• If ${\displaystyle x\neq y}$  are two points in ${\displaystyle X}$ , then there is a holomorphic function ${\displaystyle f\in {\mathcal {O}}(X)}$ , such that ${\displaystyle f(x)\neq f(y)}$ .
• For every point ${\displaystyle x\in X}$ , there are ${\displaystyle n}$  holomorphic functions ${\displaystyle f_{1},...,f_{n}\in {\mathcal {O}}(X)}$ , which form a coordinate system at ${\displaystyle x}$ .

Properties and examples of Stein manifolds

• The standard complex space ${\displaystyle \mathbb {C} ^{n}}$  is a Stein manifold.
• It can be shown quite easily, that every submanifold of a Stein manifold is a Stein manifold, too.
• The embedding theorem for Stein manifolds states the following: Every Stein manifold ${\displaystyle X}$  of complex dimension ${\displaystyle n}$  can be embedded into ${\displaystyle \mathbb {C} ^{2n+1}}$ . (The proof of this theorem requires some harder analysis).

Gathering these facts, one sees, that Stein manifold is a synonym for a submanifold of complex space.

• A consequence of the embedding theorem is the following fact: a connected Riemann surface (i.e. a complex

manifold of dimension 1) is a Stein manifold if and only if it is not compact.

• Being a Stein manifold is equivalent to be a (complex) strongly pseudoconvex manifold. (The latter means, that it has a strongly pseudoconvex exhaustive function, i.e. a smooth real function ${\displaystyle \psi }$  on ${\displaystyle X}$  with ${\displaystyle i\partial {\bar {\partial }}\psi >0}$ , such that the subsets ${\displaystyle \{z\in X,\psi (z)<c\}}$  are compact in ${\displaystyle X}$  for every real number ${\displaystyle c}$ ). This question was the so-called Levi problem.

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" holomorphic functions taking values in the complex numbers. See for example Cartan's theorems A and B, relating to sheaf cohomology. The initial impetus was to have a description of the properties of the ___domain of definition of the (maximal) analytic continuation of an analytic function.

In the GAGA set of analogies, Stein manifolds correspond to affine varieties.

Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory".

Literature

Lars Hörmander, An Introduction to Complex Analysis in Several Variables, Van Nostrand (including a proof of the embedding theorem)